Find the derivative of g(z) = 4z^7 - 3z^-7 + 9z

Are you certain your teacher expects you to answer this via first principles? Seems kind of cruel and unnecessary. All I'll tell you is that the answer should be:

\(28z^6+21z^{-8}+9\)

*note: I found this using a more practical way of finding derivatives... You probably haven't learned it yet

Also keep in mind that whatever \(\left(z+h\right)^7\) expands to, you can also use it for the second term, that can be written as \(\frac{3}{z^7}\) instead.

My advice is that if this isn't for an assignment (i.e. for marks), I wouldn't worry about it moving forward.

well we haven't actually done an example like this in class. I came across this question as I was searching practice questions for finding the derivative using first principles. I just thought it was important to know these kind of questions just in case if I ever get tested on it.

Done that ^

I hope you don't take this the wrong way but I don't think I'll go ahead to do this, it'd take me like 10 pages 😆 Sorry, slimes

What I would do instead is look at the difficulty level of your textbook, and don't go beyond that

Totally understandable. using first principles for a question like this will take a very long time. I just needed an explanation on how to cancel everything out, once factoring all of g(z) = 4z^7 - 3z^-7 + 9z. There is a fraction on top of the fraction itself, like what I had showed ^^ it's hard to tell you what I mean, a lot easier if I showed you all of the steps that I did on paper.
Will do that

Yeah, that's why mathematicians developed techniques to find derivatives without having to use the "definition of a derivative". I wouldn't mind showing you and example of lesser powers, say x^2 and x^-2, but x^7 is just overkill.

sure you can attempt to do the same question except you can use a lesser exponent such as 2 or 3. that would be extremely helpful, thanks

Hi bio man, have you tried to attempt the question? sorry if sounds like I'm pushing you or anything, but just curious because it would have helped me a lot to see the full solution of this question. I have been working on other questions as well, but just without the negative exponent's because I'm just unsure of the steps.
Thanks.

Hi slimes, I haven't forgotten about your request. Just to give you a sense of how it's done, take a look at this video. It uses first principles to find the derivative of \(f\left(x\right)=\frac{1}{x}\), which is equivalent to \(f\left(x\right)=x^{-1}\)



I want you to notice the algebraic complexity required to do what you're requesting.

Hey bio_man, would you mind still showing me the full solution for this question, but instead with exponents 2/-2. I've been ignoring this question, as I was practicing other derivative questions, so now I want to come back to this one. hope that's okay, Thanks.
ALSO, I did watch the video but I still would like to see the solution.

Like I said already, no teacher is going to ask you to do this using the definition of a derivative, and here's why:





The answer is a bit off, should be + 9 instead of + 48, but that's something you can sort through. Take a good look at the amount of algebra needed, even a seasoned veteran like myself messed up a little. No offence, but I will not be answering questions on this topic moving forward

Yes it was a difficult question to solve using for first principles, but I still wanted to attempt, which I did. So sorry for making you do this type of question, as a I searched for practice questions with first principles, I was given a lot of difficult/long processed ones like this. Thank you for taking the time to do this question, I will use it to help me with other questions.

You're welcome. The reason teachers don't ask these questions is because it becomes less calculus and more algebra, and the purpose of calculus assessments is to see if you can apply calculus concepts, and not to the point that it's consumed completely by the algebra.

Good luck